Perturbative QCD fitting of the $e^+e^-$ to hadrons KEDR and BESIII data for R(s) and $\alpha_s$ determination

This paper analyzes KEDR and BESIII experimental data for the $e^+e^-$ to hadrons R-ratio below the charm threshold using perturbative QCD at various truncation orders to extract $\alpha_s(M_Z)$ values, revealing a significant dependence of the results on the order of the approximation and highlighting the importance of handling analytical continuation effects.

The Gist

Imagine the universe is a giant, complex machine, and Quantum Chromodynamics (QCD) is the instruction manual for how the smallest building blocks of matter (quarks) stick together to form protons and neutrons. The "glue" holding them together is a force carried by particles called gluons.

The strength of this glue isn't constant; it changes depending on how much energy you use to probe it. Physicists call this strength $\alpha_s$ (the strong coupling constant). To test if our instruction manual (QCD) is correct, we need to measure this strength at different energy levels and see if it matches the predictions.

This paper is like a team of mechanics (Kataev and Todyshev) trying to tune the engine of this universe by looking at data from two different car races: one run by the KEDR team in Russia and another by the BESIII team in China. They are studying what happens when electrons and positrons (matter and antimatter) smash together and turn into a shower of hadrons (particles made of quarks).

Here is the story of their investigation, broken down simply:

1. The Experiment: Smashing Particles

Think of the electron-positron collision as a high-speed crash test. When they smash, they create a burst of energy that turns into new particles. The physicists measure a ratio called $R(s)$, which is essentially asking: "Out of all the energy we put in, how much turned into new hadrons compared to how much turned into simple muons?"

This ratio is a direct reflection of the strength of the strong force ($\alpha_s$). If we know the rules of the game (the math of QCD), we can look at the crash results and calculate exactly how strong the glue is.

2. The Problem: The "Zoom Level" Issue

The math used to predict these crashes is like a map. You can draw a map with low detail (Low Order), medium detail (Next-to-Leading Order), or high detail (Next-to-Next-to-Leading Order).

• The KEDR Data: This is a very clean, well-behaved set of measurements. It's like a clear, sunny day at the race track.
• The BESIII Data: This data is a bit more chaotic. Specifically, at higher speeds (energies between 3.4 and 3.8 GeV), the BESIII measurements seem to be "drifting off the road." They are higher than what the standard math predicts.

3. The Investigation: Trying Different Maps

The authors tried to fit the experimental data to their theoretical maps at different levels of detail:

• Level 1 (NLO): A basic map.
• Level 2 (NNLO): A better map with more curves.
• Level 3 (N3LO): A very detailed map with every pothole marked.

The Twist:
When they used the basic and medium maps (NLO and NNLO), the data fit beautifully. The calculated strength of the glue ($\alpha_s$) came out to be around 0.118 to 0.122, which matches what other experiments in the universe have found.

However, when they switched to the super-detailed map (N3LO), something strange happened. The math started to go haywire. The calculated strength of the glue jumped up to 0.131.

4. The "Ghost" in the Machine: Analytical Continuation

Why did the super-detailed map fail? The authors explain this using a concept called Analytical Continuation.

Imagine you are walking in a park (the "Euclidean" world of pure math). You calculate the distance to a tree. Then, you try to apply that same calculation to a different park (the "Minkowski" world where real experiments happen).
In the real world, there are "ghosts" (mathematical terms involving $\pi^2$) that appear when you cross from the math world to the real world. These ghosts change the sign of the numbers in the equation.

• In the math world, the terms might be all positive.
• In the real world, the "ghosts" flip some of them to negative.

The authors found that at the highest level of detail (N3LO), these "ghosts" create a series of numbers that bounce up and down wildly (positive, negative, positive, negative). This makes the prediction unstable, causing the calculated strength of the glue to look artificially high.

5. The Solution: Cutting the Bad Data

The BESIII data had a specific problem: the 8 points at the highest energies were "driving off the road." The authors decided to ignore those 8 points and only look at the 6 points below the "J/$\Psi$ meson" threshold (a specific energy barrier).

When they combined the clean KEDR data with the trimmed BESIII data:

• The NLO and NNLO results were rock solid and agreed with the rest of the physics community.
• The N3LO result still tried to push the value up, but the authors argue this is likely a flaw in the "super-detailed map" method, not a flaw in nature.

The Big Takeaway

The paper concludes that:

1. The Standard Model is likely correct: If we use the "medium detail" math (NNLO), the strength of the strong force ($\alpha_s$) is consistent with everything else we know.
2. Beware of over-fitting: Trying to use the most complex math available (N3LO/N4LO) on this specific type of data might actually make things less accurate because of those "ghost" terms that flip the signs.
3. Data matters: The BESIII data at the very highest energies in this range seems to have some tension with the theory, suggesting we need to be careful about how we interpret those specific points.

In a nutshell: The authors are saying, "We looked at the crash test data. The simple and medium-level math works perfectly and tells us the glue strength is about 0.12. The super-complex math is getting confused by some mathematical ghosts and giving us a wrong answer. Let's stick to the reliable middle ground for now."

Technical Summary

1. Problem Statement

The paper addresses the determination of the strong coupling constant, $\alpha_s(M_Z)$, using experimental data from electron-positron ($e^+e^-$) annihilation into hadrons. Specifically, it focuses on the energy region below the charm quark threshold ($\sqrt{s} < 3.7$ GeV), where the number of active quark flavors is $f=3$.

The core challenges identified are:

• Data Tension: There is a known discrepancy between recent experimental data from the KEDR (Novosibirsk) and BESIII (Beijing) collaborations. BESIII data points in the 3.4–3.6 GeV range lie significantly higher than KEDR data and theoretical Next-to-Next-to-Next-to-Leading Order (N3LO) QCD predictions.
• Perturbative Stability: Previous analyses (e.g., Ref. [33]) fixed $\alpha_s$ to the world average and analyzed the fit quality. This paper aims to extract $\alpha_s$ directly from the data to test the stability of the extraction across different orders of perturbation theory (LO, NLO, NNLO, N3LO, and estimated N4LO).
• Analytical Continuation: The paper investigates the impact of transforming perturbative expansions from the Euclidean region (where calculations are typically performed) to the physical Minkowski region (where experimental data exists). This involves handling $\pi^2$-enhanced terms arising from analytical continuation.

2. Methodology

Theoretical Framework

• R-ratio Definition: The study utilizes the ratio $R(s) = \sigma_{tot}(e^+e^- \to \text{hadrons}) / \sigma_{tot}(e^+e^- \to \mu^+\mu^-)$.
• Perturbative Expansion: The authors use Fixed-Order Perturbation Theory (FOPT) in the $\overline{\text{MS}}$ scheme. They express $R(s)$ and the Euclidean Adler function $D(Q^2)$ as power series in the coupling constant $a_s = \alpha_s/\pi$.
• Analytical Continuation: A critical component is the treatment of the transition from Euclidean to Minkowski space. The authors explicitly calculate the $\pi^2$-terms ($\Delta_k$) that modify the coefficients of the $R(s)$ series starting at order $O(\alpha_s^3)$. This leads to a sign change in higher-order coefficients compared to the Euclidean expansion.
• Renormalization Group (RG): The evolution of $\alpha_s$ is governed by the QCD $\beta$-function up to N4LO. The authors solve the RG equation to determine $\Lambda_{\overline{\text{MS}}}^{(f=3)}$ and evolve it to $\Lambda_{\overline{\text{MS}}}^{(f=5)}$ to obtain $\alpha_s(M_Z)$.
• Flavor Thresholds: The transition from $f=3$ to $f=4$ and $f=5$ is handled using matching conditions at the $D^0$ ($1.86$ GeV) and $B_c$ ($5.28$ GeV) meson mass scales, including NLO, NNLO, and N3LO matching corrections.

Experimental Data and Fitting Procedure

• Datasets:
  • KEDR: 22 data points in the range $1.84 \le \sqrt{s} \le 3.72$ GeV.
  • BESIII: 14 data points in the range $2.23 \le \sqrt{s} \le 3.67$ GeV.
• Covariance Matrices:
  • For KEDR, the full covariance matrix (statistical + correlated systematic) is used as provided in the original publication.
  • For BESIII, where the correlation matrix was not explicitly given, the authors constructed it by separating uncertainties into correlated (luminosity, trigger, etc.) and uncorrelated components based on the original paper's captions.
• $\chi^2$ Minimization:
  • $\chi^2_0$: Standard minimization using the inverse covariance matrix.
  • $\chi^2_1$: A modified minimization function that introduces a free normalization parameter $\nu$. This accounts for potential overall shifts in systematic uncertainties (e.g., luminosity normalization) by scaling the experimental data. This method was previously used in high-energy fits.
• Fitting Strategy: The authors perform fits for KEDR alone, BESIII alone (both full and truncated), and a combined fit of KEDR and truncated BESIII data.

3. Key Contributions

1. Direct Extraction of $\alpha_s$: Unlike previous studies that fixed $\alpha_s$ to the PDG average, this work extracts $\alpha_s(M_Z)$ directly from low-energy $e^+e^-$ data, testing the consistency of the extraction across different perturbative orders.
2. Treatment of Analytical Continuation: The paper rigorously applies the $\pi^2$-enhanced corrections to the $R(s)$ coefficients, demonstrating how these terms alter the asymptotic behavior of the perturbative series in the Minkowski region.
3. Resolution of Data Tension: The authors demonstrate that the tension between KEDR and BESIII data is localized to the energy region near the $J/\Psi$ and $\Psi'$ resonances (above 3.4 GeV). By truncating the BESIII dataset to exclude points above the $J/\Psi$ mass, a consistent fit with KEDR data is achieved.
4. Systematic Uncertainty Handling: The use of the $\chi^2_1$ method with a free normalization parameter $\nu$ provides a robust way to handle correlated systematic errors, particularly for the BESIII dataset where such correlations were not explicitly provided.

4. Results

KEDR Data Fits

• The fits to KEDR data alone show a stable $\chi^2$ per degree of freedom.
• As the perturbative order increases from NLO to N3LO, the extracted value of $\alpha_s(M_Z)$ increases.
  • NLO: $\alpha_s(M_Z) \approx 0.1182$ (with $\chi^2_1$)
  • NNLO: $\alpha_s(M_Z) \approx 0.1223$
  • N3LO: $\alpha_s(M_Z) \approx 0.1304$
• The inclusion of estimated N4LO corrections causes a slight decrease in the N3LO result, suggesting convergence issues or the need for higher-order terms.

BESIII Data Fits

• Full Dataset: Fits to the full BESIII dataset (including points > 3.4 GeV) yield unsatisfactory results with very high $\chi^2$ values ($\sim 50/12$) and unphysical values for $\Lambda_{\overline{\text{MS}}}^{(3)}$ (e.g., $\sim 20$ MeV). This confirms the tension between BESIII data and perturbative QCD predictions in the resonance region.
• Truncated Dataset: When BESIII data is truncated to exclude points above the $J/\Psi$ mass ($\sqrt{s} < 3.1$ GeV), the fit quality improves significantly, and the results become consistent with KEDR data.

Combined Fits (KEDR + Truncated BESIII)

The most robust results come from the combined fit of 22 KEDR points and 6 truncated BESIII points:

• NLO: $\alpha_s(M_Z) = 0.1179^{+0.0051}_{-0.0069}$
• NNLO: $\alpha_s(M_Z) = 0.1221^{+0.0063}_{-0.0080}$
• N3LO: $\alpha_s(M_Z) = 0.1312^{+0.0027}_{-0.0067}$
• N4LO (Estimate): $\alpha_s(M_Z) \approx 0.1268$

Key Observation: There is a clear tendency for $\alpha_s(M_Z)$ to grow as the perturbative order increases from NLO to N3LO. This growth is attributed to the large negative contributions from the analytical continuation terms ($\pi^2$-terms) in the N3LO and N4LO coefficients, which destabilize the fixed-order expansion in the Minkowski region.

5. Significance and Conclusion

• Validation of Low-Energy QCD: The study confirms that perturbative QCD can successfully describe $e^+e^-$ annihilation data below the charm threshold, provided the data is carefully selected to avoid resonance regions and systematic uncertainties are handled via normalization parameters.
• Perturbative Series Behavior: The paper highlights the non-asymptotic nature of the fixed-order perturbative series in the Minkowski region due to analytical continuation effects. The sign changes in higher-order coefficients lead to a "running" of the extracted $\alpha_s$ value with the order of truncation.
• Implications for $\alpha_s$ Determination: The extracted NNLO value ($0.1221$) is in good agreement with other determinations from higher energy $e^+e^-$ data and the PDG average. However, the N3LO shift suggests that fixed-order perturbation theory (FOPT) may face limitations in this energy regime, potentially supporting the use of alternative approaches like Contour Improved Perturbation Theory (CIPT) or Analytic Perturbation Theory (APT), as hinted at in the discussion of $\tau$-lepton decays.
• Data Consistency: The work clarifies that the discrepancy between KEDR and BESIII is likely due to specific issues in the BESIII data above 3.4 GeV (near resonances) rather than a fundamental failure of QCD in the lower energy region.

In summary, the paper provides a rigorous re-analysis of low-energy $e^+e^-$ data, offering new precise values for $\alpha_s(M_Z)$ while critically examining the stability of perturbative QCD predictions and the handling of experimental systematic errors.